Let $f$ be a twice differentiable function, and let $f(2)=3$, $f'(2)=0$, and $f''(2)=5$. What occurs in the graph of $f$ at the point $(2,3)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(2,3)$ is a minimum point. (Choice B) B $(2,3)$ is a maximum point. (Choice C) C There's not enough information to tell.
Since $f'(2)=0$, we know that $x=2$ is a critical point. The second derivative test allows us to analyze what happens in the graph of $f$ at this point according to these three cases: If $f''(2)>0$, the graph of $f$ has a minimum point at $x=2$. If $f''(2)<0$, the graph of $f$ has a maximum point at $x=2$. If $f''(2)=0$, the test is inconclusive. [Why is this so?] We are given that $f''(2)=5>0$. Therefore, $(2,3)$ is a minimum point.